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化工應用數學. Transform Method for Solutions of Partial Differentiation Equations. 授課教師： 林佳璋. Laplace Transform. Laplace Transform. Laplace Transforms of Derivatives. Shifting Theorem. Inverse Transformation. Inverse Transformation. Evaluating B by equating coefficients of s 2.
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化工應用數學 Transform Method for Solutions of Partial Differentiation Equations 授課教師： 林佳璋
Inverse Transformation Evaluating B by equating coefficients of s2
Example Using the data given below obtain a preliminary estimate of the diameter of the tubes to be installed in a fixed bed catalytic reactor which is to be used for the synthesis of vinyl chloride from acetylene and hydrogen chloride. The tubes are to contain the mercuric chloride catalyst deposited on 2.5 mm particles of carbon and the heat of the reaction is to be employed to generate steam at 120 C for the remainder of the process. To do this, the temperature of the inside surface of the tubes should be constant at 149 C. Effective thermal conductivity of bed (kE) = 25.4 kJ/m K h Heat of reaction at bed temperature (-H) = 1.07105 kJ/kg mol Bulk density of bed () = 290 kg/m3 The rate of reaction is a function of temperature, concentration and the various adsorption coefficients, but for the preliminary estimate assume that the rate of reaction can be expressed as r=r0(1+AT) kg moles/h kg catalyst Where r0=0.12, A=0.043, and T is the temperature in degrees Kelvin above a datum of 366 K. The maximum allowable catalyst temperature to ensure a satisfactory life is 525 K ( that is, T=159 C).
Example G = flow rate in kg/h m2 of reactor section R = tube radius in m x = radial coordinate z = axial coordinate from inlet Cp = heat capacity of the gas in kJ/kg K x z G R
Laplace Transform Method for P.D.E. The Laplace transform can remove the derivatives from an O.D.E. The same technique can be used to remove all derivatives w.r.t. one independent variable from a P.D.E. provided that it has an open range. A P.D.E has two independent variables can use “the Laplace transform method” to remove one of them and yields an O.D.E.. The boundary conditions which are not used to transform the equation must themselves be transformed.
x dx Unsteady-state One-dimensional Heat Conduction Boundary condition The initial condition can use the Laplace transform method and x and t are independent variables Laplace transform second order linear O.D.E. s regards as constant
Unsteady-state One-dimensional Heat Conduction boundary condition Laplace transform and B.C. when x , T remains finite remains finite B = 0 inverse transform
x dx Unsteady-state One-dimensional Heat Conduction Boundary condition the heat is concentrated at the surface initially and x and t are independent variables Laplace transform second order linear O.D.E. s regards as constant
Unsteady-state One-dimensional Heat Conduction boundary condition Laplace transform x and t are independent variables inverse transform
x dx Unsteady-state One-dimensional Heat Conduction Boundary condition the heat is supplied at a fixed rate and x and t are independent variables Laplace transform second order linear O.D.E. s regards as constant
Unsteady-state One-dimensional Heat Conduction boundary condition Laplace transform x and t are independent variables inverse transform
Heat Conduction between Parallel Planes Consider the flow of heat between parallel planes maintained at different temperatures T0 Boundary condition x The initial condition can use the Laplace transform method T1 and x and t are independent variables Laplace transform second order linear O.D.E. s regards as constant
Heat Conduction between Parallel Planes boundary condition Laplace transform inverse transform
Symmetrical Heat Conduction between Parallel Planes Consider a wall of thickness 2L with a uniform initial temperature throughout, and let both faces be suddenly raised to the same higher temperature. Boundary condition x The initial condition can use the Laplace transform method and x and t are independent variables Laplace transform second order linear O.D.E. s regards as constant
Symmetrical Heat Conduction between Parallel Planes boundary condition Laplace transform Laplace transform
Symmetrical Heat Conduction between Parallel Planes inverse transform
Exploitation of an Oilfield An extensive shallow oilfield is to be exploited by removing product at a constant rate from one well. How will the pressure distribution in the formation vary with time? Taking a radial coordinate r measured from the base of the well system, it is known that the pressure (p) follows the normal diffusion equation in the r direction: where is the hydraulic diffusivity If the oil is removed at a constant rate q: where k is the permeability; h is the thickness of the formation; and is the coefficient of viscosity
Exploitation of an Oilfield and r and t are independent variables Laplace Transform second order O.D.E Modified Bessel’s equation
Exploitation of an Oilfield boundary condition Laplace transform inverse transform
Gas Absorption in a Falling Film A wetted wall column is to be used for the absorption of gas (A). The gas is consumed in the liquid phase by a pseudo first order reaction in terms of the gas concentration. Develop expressions giving the point absorption rate and effective penetration depth as a function of distance from the liquid inlet. v z x Although it is poor assumption which depends upon inlet concentrations, the parabolic velocity distribution v(x) will be assumed to exist right at the inlet and remain unchanged throughout the length of the column.
Gas Absorption in a Falling Film Laplace transform
Gas Absorption in a Falling Film Point absorption rate per unit area is given by
Gas Absorption in a Falling Film Penetration depth is given by
Restriction on Use of Laplace Transform The problem must be of initial value type. The dependent variable and its derivative remain finite as the transformed variable tends to infinity. The Laplace transform should be tried whenever a variable has an open range and the method of separation should be used in all other cases. There are many P.D.E.s which cannot be solved by either method, and the numerical methods are recommended.
Other Transforms Fourier sine transform limitation 1.The variable to be transformation must be an open range. 2.A type 1 boundary condition must be given at the lower limit. 3.The dependent variable and its derivative must vanish as x. 4.All derivatives to be transformed must be of even order.
Other Transforms Hankel transform Mellin transform
